Would researchers understand that researcher's decision?
You seem to be imagining the other researchers as being much more interested in the personal life and preferences of someone they don’t know than they actually are. In truth, the researchers would not “understand”, simply because they have no particular interest in why someone chooses to go or not to go to some city/country/university. They won’t spend any time even thinking about it. It is universally understood that people have personal preferences and that moving to another geographical location is a big decision that impacts all aspects of the person’s life. Beyond that, no one cares.
Would the researcher have to go in order to continue their career?
Absolutely not. Of course, if they don’t go, they are passing over a professional opportunity. There is no guarantee that an equally good one will present itself in the next application round. But I have a feeling the researcher has already considered this aspect of the situation…
Would it look bad if letters of recommendation said, "He/she got an offer in _, but turned it down. So he/she is applying again."?
No sensible person would ever think of writing such a thing in a letter of recommendation. It is a completely irrelevant fact that has no bearing on the candidate’s application or on any kind of information a letter of recommendation is traditionally expected to provide.
In the event that someone does write such a thing in a letter, no sensible person would find such a fact relevant to the strength of the application or would be inclined to let it influence their judgment.